Sunday, November 25, 2012

Before we start, I'd like to thank listeners JMS and Daniel54600, who posted nice reviews recently on iTunes. Remember, if you like the podcast, seeing good reviews does help motivate me to record the next episode!

Anyway, on to today's topic. Recently I read online about a research study that was published this past summer in the Proceedings of the US National Academy of Sciences, called "Heavy Use Of Equations Impedes Communications By Biologists", by Tim Fawcett and Andrew Higginson. The authors analyzed a large number of recent biology papers, and tried to relate the number of equations in each paper to the citation count, or number of later papers that referenced it. They concluded that the more equations you have, the fewer people will later read and use your paper: each increase of 1 equation per page caused a 28% penalty in citations. So, does this really mean that scientists are afraid of math?

As you would expect, the Fawcett-Higginson paper led to quite a bit of discussion in the blogosphere. Do scientists really hate equations? One caveat that was often pointed out is the fact that theoretical papers with lots of equations tend to be more specialized, which inherently leads to lower citation rates. You can probably think of many other factors in the audiences and targets of papers which might have led to the reported results. On the other hand, there are also numerous articles supporting the implication that biologists don't like math in their papers, one pointing out that some Ph.D.s in the life sciences never had to take a math class beyond calculus. Going through the equations can be difficult-- one blogger linked in the show notes suggests that reading papers should be accompanied by "Eye tracking and mild electroshock therapy. If scientists skim over pages of equations or stare into space for too long while reading a technical paper, they get a gentle jolt of electricity to bring them back to the important equations at hand." A bit of hyperbole, but a good way to highlight the amount of mental discipline it takes to follow a detailed series of equations in a paper.

Reading about this paper brought to mind memories of my graduate studies in computer science, back in the early 90s. One day I had come to my advisor with a proposal for my dissertation topic, involving efficient checkpointing methods for parallel programming systems. After looking over my proposal, he looked up at me and said, "It's a good topic, but can you work in more equations?" I was a little confused, and asked where he thought I had skipped a needed equation. "Nowhere specific, it's just that more equations are expected." Needless to say, that was one of the more frustrating conversations of my aborted academic career.

I think what most discussions of the paper are missing is to ask the question: what is the role of an equation in a paper? Is it a quasi-mystical invocation, as my advisor seemed to think, adding credibility to the thesis regardless of its content? Is it an ego-boosting method for the author, allowing him to claim a deep understanding that is inaccessible to the casual reader? Obviously, neither of these is a very good answer. My answer would be that in a science or engineering paper equations perform a very important role: they allow you to rigorously prove that some result is a consequence of your assumptions, definitions, and experimental work. Mathematics is what you use to start with precise definitions and assumptions, and show their logical consequences. When you can derive a new equation to describe a concept in a rigorous, universally applicable way, that is one of the most powerful results possible in a piece of scientific work.

But do scientists and engineers usually work by spending their days deriving series of equations? Actually, before there is an equation, there is usually some kind of intuition about how the world works. And it is refinining and clarifying these intuitions and experiments in a precise way that lead to the mathematical results. Often when writing a paper, proud of the difficult and rigorous work it took to derive the equations based on your original theories, it is easy to get caught in the trap of wanting to put those equations up front as the primary focus of your communication. It's not the fault of the authors that they make this mistake, as it has been programmed in us starting as early as high school geometry. Were you in a class where you were presented a series of proofs that seemingly sprang in ordered, fully understood form from Euclid's mind? I would bet that each of his results certainly came from many hours of doodling and experimenting with different pictures and measurements. How many triangles do you think the ancient Greeks drew before they were ready to prove the Pythagorean Theorem?

When you're writing a paper in any scientific or engineering discipline, you need to keep in mind that your primary job is to communicate with your audience-- to enable the reader to understand the ideas in your paper. If you have performed some insightful math to verify the surprising and more general consequences of your initial intuitions, it is important to tell your readers about that, BUT also important to help them understand your original intutions that led them to the equations. The first reading of a paper by an individual reader will nearly always be a casual attempt to get the basic idea; only after they have been convinced intuitively of the value of the new contribution will they take the time and mental energy to follow all the details. I've sat through way too many lectures at conferences that presented a dense series of derivations and equations one by one, rather than describing at a high level what they are talking about. The results were probably really good, but without any intuition to grab on to, it's nearly impossible to stare at a long series of equations on the screen or in a paper without zoning out.

So I wouldn't take the Fawcett-Higginson result as a statement against equations-- but as a call for theoreticians to keep their audience in mind when trying to describe their results. If describing the intuition first and then putting the detailed equations in an appendix makes the paper more understandable, they should not be afraid to do it. It doesn't detract from the math to provide the audience with an intution about what inspired it, and often can make it much more likely for the work to ultimately be understood and built upon. And isn't that the real point of a scientific or engineering paper?

Sunday, October 28, 2012

Back in episode 128, you may have heard
me rightly make fun of New Age numerologists for their propensity to
draw great significance into adding up the digits of a number,
ignoring the various deep and interesting aspects of numerical
structure. I still reserve the right to make fun of them for this,
as it's generally pretty silly to prognosticate the future that way.
But, being a thoughtful Number 7 personality, it occurred to me that
it might be worth talking about the legitimate uses for summing the
digits of a number. While it doesn't have quite the universal
significance taught by the numerologists, adding the digits of a
number is a legitimate mathematical operation, known as the “Digital
Root”, and does have real applications.

To start with, let's take a closer look
at what you get when you add a number's digits, repeating the process
if the result is multi-digit, until you have digested the original
number down to a 1-digit value. At first you might think the result
is totally arbitrary, but as with many things in math, if we look a
little closer we can find a surprising pattern. Try adding the
digits of a few examples, and you may start to notice that the
digital root is congruent modulo 9 to the original number. In other
words, if you take the number you started with, divide it by 9, and
look at the remainder, you'll get the same value as you did by this
digit-summing process. (Note that for this purpose, as is standard
in modular arithmetic and bargain sales prices, we are treating 9 as
equivalent to 0.) For example, look at 56. Since 5+6=11. and
1+1=2, its digital root is 2. And it's also 2 more than the closest
multiple of 9, 54. How did that happen?

It's actually pretty easy to prove that
this is always the case. Suppose we have a 4 digit number abcd.
Remember that in our place-based number system, the value of this
number is equivalent to 1000a + 100b + 10c + d. Let's rearrange
this a bit, and write it as 999a + 99b + 9c + a +b+c+d. When you
write it this way, you can see that the original number is equal to
some multiple of 9 plus the sum of the digits. If the sum of the
digits is itself multi-digit, you can repeat the process for another
iteration or two. In any case, you can see that our result is
proven: this sum really is congruent mod 9 to the starting number.

If you're lucky enough to have gone to
elementary school before the age of calculator addiction, you may
vaguely recall one powerful use for this property of digital roots:
the old trick of “casting out 9s”, to check if the result of an
arithmetic operation you did by hand is accurate. This takes
advantage of the fact that the mod 9 value is preserved by addition,
subtraction, multiplication, and division: so if P is congruent to
3, and Q is congruent to 2, then P+Q is congruent to 5. So if you
find the digital root of P and add it to the digital root of Q, the
sum should match the digital root of your result for P+Q-- if it
didn't, you made a mistake somewhere. Be careful though-- about 1/9
of your mistakes will probably result in an error that is congruent
mod 9, and will be missed by this method. The term “casting out
9s” comes from the fact that you can take a shortcut and cross out
any sets of digits that sum to 9 before adding up the digital roots,
without changing the results.

There are also many cases where certain
types of numbers leave “tracks” in the digital roots that can be
used to quickly eliminate candidates that might or might not be
numbers of a certain type. As the simplest example, numbers
divisible by 9 are identifiable by their digital root of 9. Related
to this is that numbers divisible by 3 always have digital roots of
3, 6, or 9. All squares have a digital root of 1, 4, 7, or 9, and
cubes have a digital root of 1, 8, or 9. On the wikipedia page you
can see the patterns for other cases such as perfect numbers, prime
numbers, twin primes, factorials, and more.

A cooler application of digital roots
is the figure known as the Vedic Square, an ancient Indian variant of
the traditional 9x9 multiplication table, with the digital roots in
each square instead of the product. So for example, in row 7 and
column 8, instead of displaying the product 56, we would write 2,
since 5+6 = 11 and 1+1 = 2. I know, you're probably thinking to
yourself, “Since when are multiplication tables cool?” But try
drawing one, and then coloring in all the squares with a particular
set of numbers-- color in all the 1s, or all the 2s, or all the 4s
and 5s, etc. You will find that you generate intricate symmetrical
patterns, which you can see at some links in the show notes if you're
too lazy to start sketching out squares yourself. Such patterns
have been observed since ancient times and have been used as the
foundations of various forms of abstract Islamic art. I also found
one web page claiming the Sistine Chapel, the I Ching, and the game
of chess were all influenced by Vedic Square patterns, but these
sound like a bit of a stretch.

We should also keep in mind that the
concept of a digital root is closely tied with the fact that our
number system is Base 10. If, for example, we decided to use a Base
12 number system, then the digital roots of any number would be
congruent modulo 11 rather than modulo 9 to it. The process of
casting out 9s would be replaced by casting out 11s. And we would
be able to generate an 11x11 Vedic square that would be totally
different from our base-10 9x9 one. Next time you're in the mood to
doodle some art & feeling uninspired, just choose an abitrary
base and draw a Vedic square of your own, and you'll generate a new
fresh set of visually interesting patterns.

Sunday, August 5, 2012

Before we start, I'd like to thank listener DeeTeeEnn, who posted another nice review on iTunes. Remember, you can have your name immortalized in podcast form as well, if you follow Dee's lead & post a review of your own. Anyway, on to our main topic.

You may have been amused or disgusted by the recent release of a cinematic bomb known as "Abraham Lincoln, Vampire Hunter". Kind of ridiculous to take an accomplished historical figure and use him as a prop in an adventure totally unrelated to his accomplishments, isn't it? But perhaps the silliest thing about this idea is that there are plenty of real-life unexpected juxtapositions that could have been mined to create much better, and less repulsive, movies. For example, suppose you saw a marquee with the title "Isaac Newton, Supercop." What would you think? Believe it or not, such a movie might very well be a historically accurate documentary. Isaac Newton, the founder of modern physics and co-inventor of calculus, was also a pioneer in many aspects of modern law enforcement, and an amazingly successful detective.

Newton's unexpected detective career began when he was appointed Warden of the Royal Mint in 1696. If you're like me, you may have read about this in the last line of some biographical article on Newton, and assumed this was a symbolic office or some kind of semi-retirement. In the past many in this post had been influential nobles who got there through connections, and didn't care about the job-- but Newton was meticulous in everything he did, and the nation was in a monetary crisis. One estimate was that 20% of the coinage was counterfeit, and low confidence in the value of money caused a resurgence of medieval-style barter. In order to restore trust in the monetary system, a great recoining was planned, where new coins would be issued that had guaranteed value and authenticity. The first thing Newton did was what you might have guessed: he carefully managed the manufacturing processes involved in recoinage, performing some of the first time-motion studies and making the mint more efficient than it had ever been, increasing its output by a factor of 10.

But taking a wider view of the problem, Netwon realized that the new coins would not solve the nation's monetary problems on its own: after they were issued, the state needed to defend their integrity. Reading the details of his job description, he discovered that he was the primary official entrusted with catching counterfeiters. At first he tried to get out of this duty, but once he realized he was stuck with it, he dove into it with the same skill he devoted to his other pursuits. Looking around and studying the situation, he figured out a few things. Creating counterfeit money was a crime that always involved multiple people: it required a location to produce the coins, accomplices to acquire the raw materials and put the coins into circulation afterwards, and complicit neighbors who would not question the noise and smoke. But anyone who you might catch red-handed with a counterfeit coin in the street would be at best a small player: getting to the source of the problem required human intelligence, direct testimony from people involved in coining conspiracies or close to them. So he recruited a network of undercover agents and informers who would hang out among the seedy locales where these criminals might be found, bringing him information about counterfeiting plots as they were being hatched. In some cases Newton himself even frequented such locations to get a feel for the atmosphere in which such conspiracies occurred. Learning from his informers, he carefully waited until he had enough independent evidence, and only then took the criminals into custody. He carefully interrogated them with methods that would generally not be too far out of place in an episode of Law and Order: after questioning them in detail about their crimes and their accomplices, he would record the results in writing and ask the subject to sign and verify that the recorded information was accurate. He conducted over 200 cross-examinations, and managed to convict 28 counterfeiters.

His most famous case was the one of William Chaloner, a sometimes successful counterfeiter who was a little too audacious for his own good. Chaloner had made a profitable career both of creating fake coinage, and of tricking others into counterfeiting plots so he could turn them in to the government and earn a reward. This techique of playing both sides of the fence worked for him for a number of years, his so-called "service to the Crown" saving him on the rare occasions when he was caught commiting crimes. When the recoinage was announced in 1696, he began scheming to use it to his advantage. He made some personal connections and managed to make a presentation to Parliament on the various ways of manufacturing false coins, which he knew well from experience, attempting to make the case that his extensive knowledge made him the perfect candidate to supervise coin production, superior to the current Warden of the Mint. If it had worked, Chaloner would have been able to counterfeit coins right at the source, from within the mint-- no doubt leading to a fortune in illicit profits. This was a massive strategic blunder on his part, though, since it brought him to Newton's attention, and Newton then took a closer look at Chaloner's colorful career. Using his skills in human intelligence, he soon connected the various dots of Chaloner's life, confirming his supicion that the supposed public servant was actually a professional criminal. Soon he was able to neutralize Chaloner's deceptive claims and convict him of his crimes.

While we might argue that these law enforcement activites are pretty trivial in comparison to his mathematical and physical contributions, it would be foolish to underestimate the importance of England's economy and monetary system at this critical time in history. You can learn a lot more about Newton's career at the Mint and his battle with Chaloner in the book "Newton and the Counterfeiter" by Thomas Levenson, which is where I first learned of this story. So next time you take some time from your day to ponder Isaac Newton's accomplishments, which you should really be doing pretty often given all that he did accomplish, don't forget that on top of the physics and the math, you should also think about the law enforcement. And next time you are having lunch with your favorite Hollywod producer, be sure to pitch Isaac Newton as an ideal subject for the next historical action thriller.

Wednesday, March 14, 2012

Nicholas Bourbaki was an influential 20th-century mathematician who published a series of texts begnning in the 1930s, trying to rigourously describe the major core areas of mathematics based on the foundations of set theory. Many of his books became standard references in their fields. Among the Bourbaki contributions that have become familiar to modern math students are the use of the slashed zero to represent an empty set, and the terms injective, surjective, and bijective. The ill-fated New Math movement in education, which I discussed back in podcast 145 (http://mathmutation.blogspot.com/2011/12/145-why-johnny-couldnt-add.html), was also largely inspired by Bourbaki ideas. But if you try to find out about the life of Bourbaki himself, you will be in for a bit of a surprise: Nicholas Bourbaki did not actually exist.

Actually, to be more precise, Bourbaki was not an individual, but a secretive society of mathematicians established by Andre Weil, Henri Cartan, and other young mathematicians in Paris in the 1930s. They originally got together because they believed there was a serious gap in the available mathematics texts of the time, a problem magnified by the fact that a generation of potential leaders in the field had been wiped out by World War I. They hoped to re-establish the rigorous foundations of mathematics, while at the same time to provide a standard series of reference works. While secret societies tend to sound sinister in concept, their secrecy actually stemmed from rather noble motives: they wanted to ensure that any works they produced would be judged on the basis of their content, not on factors related to personal egos. So they created the ficticious persona of Nicholas Bourbaki, member of the Royal Academy of Poldavia, and agreed that he would be credited as the author of all books they produced.

The members of the Bourbaki group enjoyed having fun with the semi-secret nature of their small club. To anyone who asked, they would claim that Bourbaki was a real person of their acquaintance, and they even printed up a set of mathematical-pun-laden wedding invitations from Bourbaki's daughter, to show to anyone who doubted his authenticity. According to the invitation, "The trivial isomorphism (aka the sacrament of matrimony) will be given to them by P. Adic, of the Diophantine Order, at the Principal Cohomology of the Universal Variety, the 3 Cartember, year VI, at the usual hour.". You wouldn't think that would fool too many people-- but this prank backfired horribly during World War II, when Andre Weil (who had fled to Finland) was arrested on suspicion of spying. This apparently encoded letter from a strange foreign contact was considered a key piece of evidence. Weil was sentenced to death, though at the last minute a friend with government contacts managed to intervene and get him pardoned.

You would think they would be done with practical jokes after that, but the Bourbaki group tried to continue the whimsical spirit, perhaps a welcome break from the incredbly serious work they were attempting. In the late 1940s, American mathematician Ralph Boas was contacted by the Encyclopedia Britannica to assist with an article on modern mathematics, and since he knew Weil, he mentioned in his article that Bourbaki was actually a pseudonym for a collaboration rather than an actual person. Boas and the Encylopedia then received a letter, claiming to have been written by Bourbaki from an ashram in the Himalayas, asking "You miserable worm, how dare you say that I do not exist?" A series of letters went back and forth on the topic, though the encyclopedia editors were eventually convinced of the truth. But then Bourbaki members began to spread a new rumor, that Ralph Boas did not actually exist, and was actually a collective pseudonym for a group of American mathematicians!

The Bourbaki group lasted for several generations, producing a series of comprehensive texts on the foundations of modern mathematics. While they made many lasting contributions, they were also criticized for focusing too narrowly on foundations, omitting areas relevant to widely applicable topics such as logic and mathematical physics. They also had a strange aversion to pictures and illustrations, which meant that although some of their books were very useful as references, they ironically were not very good as textbooks. The group gradually declined after the 1970s, with their last major text, "Spectral Theory", being published in 1983. The decline may have been hastened by a long legal battle with their publishing company over royalties and translation rights: Pierre Cartier, one of the members during the final productive period, described the result as "both parties lost and the lawyer got rich." In the show notes, you can find a link to a 1997 interview with Cartier, who shares some fascinating thoughts on the rise and fall of the Bourbaki group. Aside from the legal issues, one of the other major reasons he gave for Bourbaki's decline was that they had successfully achieved their objectives. As Cartier describes it, "In a given science there are times when you have to take all the existing material and create a unified terminology, unified standards, and train people in a unified style. The purpose of mathematics, in the fifties and sixties, was that, to create a new era of normal science. Now we are again at the beginning of a new revolution."

Sunday, February 19, 2012

Before we get started, I'd like to thank listeners Foxy McLovin and The Devonian Kid, who recently posted nice reviews on iTunes. Remember, you too can experience the thrill of having your name, or bizarre iTunes nickname, mentioned on Math Mutation by posting a review. Or you can send a donation to your favorite charity in honor of Math Mutation, and email me to tell me about it. Anyway, on to today's topic.

During this Valentine's week, it occurred to me that there was not yet a Math Mutation episode focusing on amicable numbers, said to be numbers that represent friendship and love. Yes, there are such things. And yes, math geeks do occasionally experience such emotions. So what exactly are amicable numbers?

To start with, let's review the concept of a perfect number. A perfect number, like 6 or 28, is precisely equal to the sum of its factors. So 6, for example, is perfect because its factors are 1, 2, and 3, and 1 + 2 + 3 = 6. The concept of amicable numbers is related to perfect numbers, except that they are pairs of numbers, such that each is the sum of the other's factors. The smallest pair of amicable numbers is 220 and 284. The factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, which add up to 284. And the factors of 284 are 1, 2, 4, 71 and 142, which add up to 220. If you stretch your sense of analogy a but, you can kind of see how the way in which two numbers are unexpectedly intertwined might suggest love and friendship.

The pair 220 and 284 was known to the ancient Greeks. But as you can probably guess, amicable numbers are relatively rare, and can be tricky to find. In 850 A.D, Arabic mathematician Thabit ibn Kurrah discovered if n > 1 and each of p = 3*2^n-1-1, q = 3*2^n-1, and r = 9*2^2n-1-1 are prime, then (2^n)pq and (2^n)r are amicable numbers. This was a nice advance, but finding numbers that meet the preconditions of this formula wasn't very easy-- several centuries later, it led to discovery of the pairs 17,296 and 18,416, resulting from n=4, and 9,363,584 and 9,437,056, which you get from n=7. Euler later published a list of 64 new amicable pairs, but he actually made some mistakes, and the list had 2 bogus entries, which was not discovered until the 20th century. But most surprisingly, the second-lowest pair of amicable numbers, 1184 and 1210, was not discovered until 1866, by a 16-year old Italian boy named Nicolo Paganini. With the help of modern computers, millions of amicable pairs have been discovered, though the conjecture that there are an inifnite number of them has not yet been proven.

Amicable numbers have been thought to symbolize love and friendship since ancient times. It is said that when Pythagoras was asked to define a friend, his definition was "one who is the other I, such as are 220 and 284." In Genesis verse 32:14, Jacob gave his brother 220 sheep when he feared he was planning to kill him, and some Torah scholars interpret this as due to its status as an amicable number, to reinforce their brotherly love. I think this last interpretation is a bit of a stretch though-- he also gave him 30 camels, 40 cows + 10 bulls, and 30 donkeys, which don't seem to have anything to do with amicable numbers. And to get the number 284 to appear somehow, scholars have interpreted one partial word in the verse as a Hebrew encoding of a number, which really seems to be reaching.

According to one online site called "Renassance Astrology", you can make a magical love talisman by the following method: "Make two images, and put one of the Fortunes at the ascendant and the Moon in Taurus conjunct Venus. Write on one image a number ... for 220 in the proper place, and on the other image write the same kind of figure for 284 in the proper place. Join the two figures together in an embrace, and then there will be perfect and lasting love between the two." They go on to supply further mystical powers of these numbers: "If the aforesaid numbers were carved in wood, and bread or anything else edible was sealed with them and you gave it to someone to eat, he will delight in you with a great love. If those numbers are written on your clothing, your garments cannot be taken away from you; and if you write them on banners that are put in the street to draw business, they will draw business to you." I'm thinking of writing some of these numbers on my socks, to stop them from disappearing in the dryer.

And at least some modern geeks still believe that amicable numbers represent love: with a quick online search, I was able to find a pair of half-heart pendants, one labeled 220 and the other labeled 284, for sale at a site called mathsgear.com . But math geekiness can only get you so far in a marriage-- when it comes to Valentine's day, I made the safe choice and got my wife some non-math-related flowers. Perhaps you will be more courageous on your next romantic occasion.
And this has been your math mutation for today.

Note that this podcast is intended mainly for audio consumption, so you will not see the numerous illustrations & diagrams you would find at most math sites, though these are linked in the show notes whenever possible.

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